Taylor's Theorem (See [1]} Lemma 3.1) 
For every function \(\phi :\mathbb {R}^J \rightarrow \mathbb {R}\)  with \(\alpha \) -Lipschitz gradients \(\nabla \phi \)  in the \(\ell _2-\ell _2\)  sense and for every \(x_0,x\in \mathbb {R}^J\)  we have
\(|\phi (x)-\phi (x_0)-\nabla \phi (x_0) \cdot (x-x_0)|\le \frac{\alpha }{2}||x-x_0||_2^2.\) 

